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1
Content available Water wave scattering by an infinite step in the presence of an ice-cover
Ray S., De S., Mandal B. N.
International Journal of Applied Mechanics and Engineering
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2019
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Vol. 24, no. 4
157--168
EN
The classical problem of water wave scattering by an infinite step in deep water with a free surface is extended here with an ice-cover modelled as a thin uniform elastic plate. The step exists between regions of finie and infinite depths and waves are incident either from the infinite or from the finite depth water region. Each problem is reduced to an integral equation involving the horizontal component of velocity across the cut above the step. The integral equation is solved numerically using the Galerkin approximation in terms of simple polynomial multiplied by an appropriate weight function whose form is dictated by the behaviour of the fluid velocity near the edge of the step. The reflection and transmission coefficients are obtained approximately and their numerical estimates are seen to satisfy the energy identity. These are also depicted graphically against thenon-dimensional frequency parameter for various ice-cover parameters in a number of figures. In the absencje of ice-cover, the results for the free surface are recovered.
2
Content available Edge waves over a shelf
Dolai P., Dolai D. P.
International Journal of Applied Mechanics and Engineering
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2019
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Vol. 24, no. 2
453--460
EN
The problem considered in this paper is the derivation of properties of edge waves travelling along a submerged horizontal shelf. The problem is formulated within the framework of the linearized theory of water waves and Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the dispersion relation for edge waves in terms of an integral. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for the integral and hence to derive the properties of edge waves over a shelf. The numerical results are illustrated in a table and curves are presented showing the variation of frequency of the edge waves with the width of the shelf.
3
Content available Scattering of oblique water waves by an infinite step
Dolai P., Dolai D. P.
International Journal of Applied Mechanics and Engineering
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2018
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Vol. 23, no. 2
327--338
EN
The present paper is concerned with the problem of scattering of obliquely incident surface water wave train passing over a step bottom between the regions of finite and infinite depth. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain very accurate numerical estimates for reflection and transmission coefficients. The numerical results are illustrated in tables.
4
Content available Oblique water wave diffraction by a step
Dolai P.
International Journal of Applied Mechanics and Engineering
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2017
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Vol. 22, no. 1
35--47
EN
This paper is concerned with the problem of diffraction of an obliquely incident surface water wave train on an obstacle in the form of a finite step. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for reflection and transmission coefficients which are depicted graphically. From these figures various interesting results are discussed.
5
Content available Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation
Just A., Stempień Z.
Opuscula Mathematica
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2016
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Vol. 36, no. 2
239--252
EN
Our goal is to study the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. First we consider a mathematical model of the beam equation which was obtained by S. Woinowsky-Krieger in 1950. Next we consider the Pareto optimal control problem based on this equation. Further, we describe the approximation of this system. We use the Galerkin method to approximate the solution of this control problem with respect to a spatial variable. Based on the standard finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation point of the solutions of the discrete optimal control problems exist and each of these points is the solution of the original Pareto optimal control problem.
6
Content available A Galerkin approximation method including space dimensional reduction - applied for solution of a heat conduction equation
Nakonieczny K.
Applied Computer Science
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2012
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Vol. 8, no 2
48--58
EN
A multivariate data fitting procedure, based on the Galerkin minimization method, is studied in this paper. The main idea of the developed approach consists in projecting the set of data points from the original, higher-dimensional space, onto a line section. Then, the approximation problem is solved in the resulting one-dimensional space. The elaborated recipe can be designed so that it is computationally more efficient than the schemes based on the least squares minimization. The performance of the method is studied by comparison with the least squares and the moving least squares procedures in a number of examples, including the solution of the heat diffusion equation.
7
Analysis and Galerkin approximation of some classes of nonlinear control problems
Just A.
Zeszyty Naukowe. Rozprawy Naukowe / Politechnika Łódzka
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1999
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Z. 260
3-71
EN
This paper is concerned with optimal control problems for the systems governed by nonlinear operator and operator-differential equations of first and second order, with monotone operators. We derive some results on the existence of optimal controls. Then we treat optimal control problems by Galerkin method and we prove the convergence of optimal values for approximated control problems to the one for the orginal problems. At the end of each Chapter we apply the obtained results to given examples for the nonlinear elliptic, parabolic, hyperbolic equations with homogeneous and non-homogeneous boundary conditions (Dirichlet and Neumann). The control is acting on the domain, in the boundary conditions or in the initial conditions.
PL
Praca ta jest poświecona globalnej analizie zadań sterowania optymalnego układami opisanymi równaniami operatorowymi i ewolucyjnymi pierwszego i drugiego rzędu z operatorami monotonicznymi w przetrzeniach Banacha i aproksymacji Galerkina tego typu nieliniowych problemów sterowania. Rozdział I to wstęp oraz wprowadzenie podstawowych definicji wykorzystywanych w tej pracy. W rozdziale II zostały przedstawione problemy sterowania optymalnego układem opisanym równaniem operatorowym Ay = u z operatorem monotonicznym A : V → V*, gdzie V jest rzeczywistą refleksywną przestrzenią Banacha a X* przestrzenią sprzężoną do V, bez ograniczeń na zbiór sterowań dopuszczalnych i z ograniczeniami. Sformułowane zostały warunki wystarczające istnienia sterowania optymalnego przy wypukłym, różniczkowalnym wskaźniku jakości. Omówiona została metoda aproksymacji Galerkina wyżej wspomnianych zadań sterowania i udowodniona zbieżność tej aproksymacji. Rozdział III jest poświecony sterowaniu optymalnemu układem opisanym równaniem ewolucyjnym pierwszego rzędu (wzór) z warunkiem początkowym y(0) = y°, z monotonicznym operatorem Volterry A. Sformułowane zostały warunki wystarczające istnienia sterowania optymalnego oraz przedstawiona została aproksymacja Galerkina i jej zbieżność dla tego typu problemu sterowania. Rozdział IV dotyczy sterowania optymalnego układem opisanym równaniem ewolucyjnym drugiego rzędu (wzór) z warunkami początkowymi y(0) = y°, y'(0) = y1, z monotonicznym operatorem Volterry A i liniowym operatorem B. Na koniec każdego z rozdziałów otrzymane wyniki zostały zilustrowane przykładami.
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